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Parabolic Monge-Ampère equations on Riemannian manifolds. (English) Zbl 0895.58053
The author considers on a compact Riemannian manifold \((M,g)\) the parabolic Monge-Ampère equation \[ {{\partial}\over{\partial t}}\varphi (x,t)=\log\left({{\det\Bigl(g(x)+\text{ Hess}\varphi (x,t)\Bigr)}\over{\det g(x)}}\right)-\lambda\varphi (x,t)-f(x),\quad \varphi(x,0)=\varphi_0(x), \] where \(\lambda\in\mathbb{R}\) is a parameter and \(\varphi_0,f\in C^\infty(M,\mathbb{R}).\) She shows global existence in time, independently of \(\lambda.\) Moreover, when \(\lambda >0\), she proves that \(\varphi_t=\varphi(\cdot,t)\) exponentially converges as \(t\to +\infty\) to a solution \(\varphi_\infty\) of the stationary problem and that, when in addition \(f=0\), one has the existence of \(\delta, c>0\) (depending on \(\varphi_0\) and \(| \nabla^i\varphi| _\infty\), \(i=0,1,2,3)\) such that \[ \int_M\bigl(\varphi_t -\overline{\varphi_t}\bigr)^2 d\text{ Vol}_g\leq c\exp\Bigl(-2(\mu_1+\lambda+e^{-\delta t})t\Bigr), \] where \(\overline\varphi\) denotes the mean value of \(\varphi\) and \(\mu_1\) is the first eigenvalue of the Laplacian.

MSC:
58J35 Heat and other parabolic equation methods for PDEs on manifolds
58J50 Spectral problems; spectral geometry; scattering theory on manifolds
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